MULTIPLICATIVE STRUCTURES AND THE TWISTED BAUM-CONNES ...barcenas/BACAVE.pdf · group via the assembly map. This is of course the case when this twisted Baum-Connes map is an isomorphism

MULTIPLICATIVE STRUCTURES AND THE TWISTED

BAUM-CONNES ASSEMBLY MAP

NOE BARCENAS, PAULO CARRILLO ROUSE, AND MARIO VELASQUEZ

Abstract

Using a combination of Atiyah-Segal ideas in one side and of Connes and Baum-Connes ideas in the other, we show that the Total Twisted geometric K-homologygroup of a Lie groupoid (includes Lie groups and discrete groups) admits a ringstructure (or module structure for the odd group). This group is the left hand sideof the twisted geometric Baum-Connes assembly map recently constructed in [5].For the case of proper groupoids, for which the assembly map is an isomorphism,our multiplicative structures coincide with the ones defined by Tu and Xu in [15].

1. Introduction

In recent years twisted K-theory and twisted index theory have benefited of agreat deal of interest from several groups of mathematicians and theoretical physi-cists. Besides its relations with string theory and theoretical physics in general,one of the main mathematical motivations was the series of works by Freed, Hop-kins and Teleman in which they describe a ring structure on an equivariant twistedK-theory of a group (compact connected Lie group) and in which they give a ringisomorphism with the Verlinde algebra of the group.

For discrete or non compact Lie groups it is not clear how these multiplicativestructures should be defined directly or even if they exist at all. In this paper wegive a step to try to understand these issues. Our approach is a mixture of Atiyah-Segal ideas in one side and of Connes and Baum-Connes ideas in the other. Indeed,if the group in question acts properly in a nice space then one can use a homotopytheoretical model for the twisted K-theory groups and use Atiyah-Segal ideas fordefining a product in this setting. On the other hand, following Baum-Connes ideasone might expect that the analytically defined twisted equivariant K-theory can beapproached (or assembled to be precise) by groups defined by using only properactions (the so called left hand side). The main result of this paper is to define aring structure on the left hand side of a twisted Baum-Connes assembly map associ-ated to every Lie groupoid, proper or not. We explain this below with more detailsbut before let us mention why we abruptly changed our terminology from groupsto groupoids. We have at least two big reasons for this, first, the category of Liegroupoids encodes much more that groups and group actions, many singular situ-ations can be handled using appropriate groupoids; second, our constructions andproofs are largely simplified by the use Connes deformation groupoids techniques(see explanation below).

We pass now to the explicit content of the paper. For proper groupoids onecan define the twisted K-theory groups by a generalization of Atiyah-Janich Fred-holm model for classical topological K-theory. More precisely, if G is a properLie groupoid with connected units M and P is a G-equivariant PU(H)−principalbundle over M , the Twisted G-equivariant K-theory groups of M twisted by P aredefined as the homotopy groups of the G-equivariant sections

Date: January 21, 2015.

1

2 NOE BARCENAS, PAULO CARRILLO ROUSE, AND MARIO VELASQUEZ

K−pG (M,P ) := πp

(Γ(M ; Fred(0)(P ))G, s

)where Fred(0)(P )→M is a certain bundle constructed from P with fibers an spaceof Fredholm operators, see definition 3.6 for more details. Using this suitable choiceof Fredholm bundles we follow Atiyah-Segal for defining an associative product

K−pG (M,P )×K−qG (M,P ′)→ K−(p+q)G (M,P ⊗ P ′).

These twisted K-theory groups for proper groupoids are isomorphic to the K-theoryof some C∗-algebras associated to the twisting, theorem 3.14 in [16]. In [15] theauthors constructed a ring structure on Twisted K-theory for proper groupoids (andmore generally for crossed modules) using the C∗-algebraic model and Kasparov’sKK-theory techniques. We check in section 3 below that under the mentionedisomorphism the multiplicative structures coincide.

For non necessarily proper groupoids one does not dispose of a Fredholm modelfor defining the multiplicative structure as above and even if there is a C∗-algebraicmodel for twisted K-theory it is not clear how to define this product directly. Follow-ing Baum-Connes ideas one might expect that the K-theory of the twisted algebracan be approached by K-theory groups using only proper actions.

Given a Lie groupoid G (not necessarily proper) together with G-equivariantPU(H)-principal bundle P over the units of G, the authors in [5] generalize tothe twisted case, Connes construction of the geometric K-homology group, de-noted by Kgeo

∗ (G,P ), and the construction of the geometric Baum-Connes assemblymap. The main theorem in order to prove that this group and the assembly mapare well defined is the wrong way functoriality of the pushforward constructionassociated to oriented smooth G-maps (theorem 4.2 in [5]). Given two isomor-phic G-equivariant PU(H)-bundles their associated twisted K-theory groups andtheir associated twisted geometric K-homology groups are isomorphic as well, alsothe twisted Baum-Connes map mentioned above is compatible with these isomor-phisms (theorem 6.4 in [5] gives a vast generalization of this fact). Denote byH1(G,PU(H)) the set of isomorphism classes of G-equivariant PU(H)-principalbundles1. Consider the Total twisted geometric K-homology group

KgeoTW,∗(G) :=

⊕α∈H1(G,PU(H))

Kgeo∗ (G,Pα)

where Pα is a G-equivariant PU(H)-principal bundle in the class of α. The groupKgeoTW,∗(G) is well defined up to isomorphism, there is no canonical choice for a

representative in a given isomorphism class.We want to briefly describe a product on this Total twisted geometric K-homology

group. By definition each Kgeo∗ (G,P ) is generated by cycles of the form (X,x)

where X is a G−proper co-compact manifold and x ∈ K−pG (X,PX) (where PX is

the twisting over X induced by P and where K−pG (X,PX) denotes the equivarianttwisted K-theory group associated to the action groupoid X o G, see definition3.6 for more details) and with main relation given by the pushforward maps (seedefinition 6.1 for more precisions).

Let P and Q two twistings on G. Let (X,x) with x ∈ K−pG (X,PX) and (Y, y)

with y ∈ K−qG (Y,QY ), the product looks like follows

(1.1) (X,x)·(Y, y) := (X×G0Y, π∗Xx·π∗Y y) ∈ K−p−qG (X×G0

Y, PX×G0Y ⊗QX×G0

Y )

1H1(G,PU(H)) is of course the 1st Cech cohomology group of G with values in PU(H) butfor the purpose of this paper we do not need to use it in these terms. There are canonical maps

H1(G,PU(H)) → H2(G,S1) → H3(G,Z) which in general are surjective and isomorphisms forthe case of proper groupoids, see for instance [16] for a detailed discution on this subject.

MULTIPLICATIVE STRUCTURES 3

where πX , πY stand for the respective projections from X ×G0Y to X and Y and

where the pullback is defined in section 6 below, it is easy to see that the productdescribed above is compatible with isomorphism classes of PU(H)−principal bun-dles. To prove that the definition above induces a product on the Total twistedK-homology group one needs to prove two main properties:

(i) The compatibility of the product with respect to the pushforward maps,proposition 5.18 below.

(ii) The compatibilty of the pushforward and the pullback constructions, propo-sition 6.7 below.

For the two properties above the use of the groupoid language becomes veryuseful. First of all the construction of the pushforward maps can be completely re-alized in the Fredholm picture by using Connes deformation groupoids, and henceadapting to this model the main results and constructions from [5] for the case ofproper groupoids, we explain this in section 5. Second, the proofs become concep-tually very simple, for example to prove the first property above amounts to checkthat the morphisms induced by restriction are compatible with the product. Soeven if one is only interested in the group case (Lie or discrete for instance) the useof deformation groupoids gives a unified way to construct the pushforwrd maps, toprove their functoriality and to prove their compatibility with the product.

Our main theorem is the following one:

Theorem 1.2. For any Lie groupoid, the product described above induces

• a ring structure on the even Total twisted geometric K-homology groupKgeoTW,0(G), and

• a KgeoTW,0(G)-module structure on the odd Total twisted geometric K-homology

group KgeoTW,1(G).

Consider, for any Lie groupoid G with an equivariant PU(H)-prinicipal bundleP , the twisted K-theory group K∗(C

∗r (G,P )), for ∗ = 0, 1. We can consider as well,

the Total Twisted K-theory group K∗TW (G) :=⊕

α∈H1(G,PU(H))K∗(C∗r (G,Pα))

and the associated Baum-Connes assembly map

(1.3) KgeoTW,∗(G)

µTW// K∗TW (G)

given in each component by the Baum-Connes assembly map

(1.4) Kgeo∗ (G,P )

µTW// K∗(C

∗r (G,P ))

constructed in [5].By the theorem above one could expect to transpose the ring structure (resp.

module structure for the odd case) to the even (resp. odd) Total Twisted K-theorygroup via the assembly map. This is of course the case when this twisted Baum-Connes map is an isomorphism. After the discussion in the last section of [5] onemight expect that this occurs whenever the untwisted Baum-Connes conjecture isverified. Another interesting question would be if it is possible to construct directlythese multiplicative structures on the Total twisted K-theory groups such that theassembly map is a ring/module isomorphism. These questions will be discussedelsewhere.

Aknowledgments. The first author thanks the support of a DGAPA researchgrant. The third Author thanks the support of a UNAM Postdoctoral Fellowship.

The first and third author thank Unversite Toulouse III Paul Sabatier, as well asthe Laboratoire International Solomon Lefschetz (LAISLA) and the ANR projectKIND for support during a visit to Toulouse. The third author thanks the Max


Planck Institut of Mathematics at Bonn for giving him excellent conditions for workout part of this project.

2. Preliminaries on groupoids

In this section, we review the notion of twistings on Lie groupoids and discusssome examples which appear in this paper. Let us recall what a groupoid is:

Definition 2.1. A groupoid consists of the following data: two sets G and M , andmaps

(1) s, r : G→M called the source map and target map respectively,(2) m : G(2) → G called the product map (where G(2) = {(γ, η) ∈ G × G :

s(γ) = r(η)}),together with two additional maps, u : M → G (the unit map) and i : G→ G (theinverse map), such that, if we denote m(γ, η) = γ · η, u(x) = x and i(γ) = γ−1, wehave

(i) r(γ · η) = r(γ) and s(γ · η) = s(η).(ii) γ · (η · δ) = (γ · η) · δ, ∀γ, η, δ ∈ G whenever this makes sense.(iii) γ · x = γ and x · η = η, ∀γ, η ∈ G with s(γ) = x and r(η) = x.(iv) γ · γ−1 = u(r(γ)) and γ−1 · γ = u(s(γ)), ∀γ ∈ G.

For simplicity, we denote a groupoid by G⇒M .

In this paper we will only deal with Lie groupoids, that is, a groupoid in whichG and M are smooth manifolds, and s, r,m, u are smooth maps (with s and rsubmersions).

2.1. The Hilsum-Skandalis category. Lie groupoids form a category with strictmorphisms of groupoids. It is now a well-established fact in Lie groupoid’s theorythat the right category to consider is the one in which Morita equivalences corre-spond precisely to isomorphisms. We review some basic definitions and propertiesof generalized morphisms between Lie groupoids, see [16] section 2.1, or [8, 13, 11]for more detailed discussions.

Definition 2.2 (Generalized homomorphisms). Let G⇒M and H ⇒M ′ be twoLie groupoids. A generalized groupoid morphism, also called a Hilsum-Skandalismorphism, from H to G is given by the isomorphism class of a principal G-bundleover H, that is, a right principal G-bundle over M ′ which is also a left H-bundleover M such that the the right G-action and the left H-action commute, formallydenoted by

f : H // G

or by

H

��

Pf

~~~~

G

��

M ′ M.

if we want to emphasize the bi-bundle Pf involved.

As the name suggests, generalized morphism generalizes the notion of strictmorphisms and can be composed. Indeed, if P and P ′ give generalized morphismsfrom H to G and from G to L respectively, then

P ×G P ′ := P ×M P ′/(p, p′) ∼ (p · γ, γ−1 · p′)


gives a generalized morphism from H to L. Consider the category GrpdHS withobjects Lie groupoids and morphisms given by generalized morphisms. There is afunctor

(2.3) Grpd −→ GrpdHS

where Grpd is the strict category of groupoids.

Definition 2.4 (Morita equivalent groupoids). Two groupoids are called Moritaequivalent if they are isomorphic in GrpdHS .

We list here a few examples of Morita equivalence groupoids which will be usedin this paper.

Example 2.5 (Pullback groupoid). Let G⇒M be a Lie groupoid and let φ : M →M be a map such that t ◦ pr2 : M ×M G→M is a submersion (for instance if φ isa submersion), then the pullback groupoid φ∗G := M ×M G×MM ⇒M is Moritaequivalent to G, the strict morphism φ∗G → G being a generalized isomorphism.For more details on this example the reader can see [11] examples 5.10(4).

Example 2.6 (Discrete groups). Let Γ be a discret group. Let M be a manifoldtogether with a generalized morphism

M −−− > Γ

(in this case this is equivalent a continuous map M → BΓ) given by a Γ-principal

bundle M → M over M (i.e., a Γ-covering). Consider the (Connes-Moscovici)groupoid

M ×Γ M ⇒M

where M ×Γ M := M × M/4Γ and with structural maps s(x, y) = y, t(x, y) = xand product

(x, y) · (y, z) := (x, z).

The groupoids M ×Γ M ⇒M and Γ ⇒ {e} are Morita equivalent.

2.2. Twistings on Lie groupoids. In this paper, we are only going to considerPU(H)-twistings on Lie groupoids where H is an infinite dimensional, complex andseparable Hilbert space, and PU(H) is the projective unitary group PU(H) withthe topology induced by the norm topology on the unitary group U(H).

Definition 2.7. A twisting α on a Lie groupoid G⇒M is given by a generalizedmorphism

α : G // PU(H).

Here PU(H) is viewed as a Lie groupoid with the unit space {e}.So a twisting on a Lie groupoid G is given by a locally trivial right principal

PU(H)-bundle Pα over G.

Remark 2.8. The definition of generalized morphisms given in the last subsectionwas for two Lie groupoids. The group PU(H) it is not a finte dimensional Lie groupbut it makes perfectly sense to speak of generalized morphisms from Lie groupoidsto this infinite dimensional groupoid following exactly the same definition.

Example 2.9. For a list of various twistings on some standard groupoids seeexample 1.8 in [6]. Here we will only a few basic examples.

(i) (Twisting on manifolds) Let X be a C∞-manifold. We can consider theLie groupoid X ⇒ X where every morphism is the identity over X. Atwisting on X is given by a locally trivial principal PU(H)-bundle over X.In particular, the restriction of a twisting α on a Lie groupoid G⇒M toits unit M defines a twisting α0 on the manifold M .


(ii) (Orientation twisting) Let X be a manifold with an oriented real vectorbundle E. The bundle E → X defines a natural generalized morphism

X // SO(n).

Note that the fundamental unitary representation of Spinc(n) gives riseto a commutative diagram of Lie group homomorphisms

Spinc(n)

��

// U(C2n)

��

SO(n) // PU(C2n).

With a choice of inclusion C2n into a Hilbert space H, we have a canonicaltwisting, called the orientation twisting, denoted by

βE : X // PU(H).

(iii) (Pull-back twisting) Given a twisting α on G and for any generalized ho-momorphism φ : H → G, there is a pull-back twisting

φ∗α : H // PU(H)

defined by the composition of φ and α. In particular, for a continuousmap φ : X → Y , a twisting α on Y gives a pull-back twisting φ∗α on X.The principal PU(H)-bundle over X defines by φ∗α is the pull-back of theprincipal PU(H)-bundle on Y associated to α.

(iv) (Twisting on fiber product groupoid) Let Np→ M be a submersion.

We consider the fiber product N ×M N := {(n, n′) ∈ N × N : p(n) =p(n′)},which is a manifold because p is a submersion. We can then takethe groupoid

N ×M N ⇒ N

which is a subgroupoid of the pair groupoid N ×N ⇒ N . Note that thisgroupoid is in fact Morita equivalent to the groupoid M ⇒M . A twistingon N ×M N ⇒ N is given by a pull-back twisting from a twisting on M .

(v) (Twisting on a Lie group) By definition a twisting on a Lie group G is aprojective representation

Gα−→ PU(H).

2.3. Deformation groupoids. One of our main tools will be the use of defor-mation groupoids. In this section, we review the notion of Connes’ deformationgroupoids from the deformation to the normal cone point of view.

Deformation to the normal coneLet M be a C∞-manifold and X ⊂M be a C∞-submanifold. We denote by NM

X

the normal bundle to X in M . We define the following set

DMX :=(NMX × 0

)⊔(M × R∗) .(2.10)

The purpose of this section is to recall how to define a C∞-structure in DMX . Thisis more or less classical, for example it was extensively used in [8].

Let us first consider the case where M = Rp × Rq and X = Rp × {0} ( herewe identify X canonically with Rp). We denote by q = n − p and by Dnp for DRn

Rpas above. In this case we have that Dnp = Rp × Rq × R (as a set). Consider thebijection ψ : Rp × Rq × R→ Dnp given by

(2.11) ψ(x, ξ, t) =

{(x, ξ, 0) if t = 0(x, tξ, t) if t 6= 0


whose inverse is given explicitly by

ψ−1(x, ξ, t) =

{(x, ξ, 0) if t = 0(x, 1

t ξ, t) if t 6= 0

We can consider the C∞-structure on Dnp induced by this bijection.We pass now to the general case. A local chart (U , φ) of M at x is said to be a

X-slice if

1) U is an open neighbourhood of x in M and φ : U → U ⊂ Rp × Rq is adiffeomorphsim such that φ(x) = (0, 0).

2) Setting V = U ∩ (Rp × {0}), then φ−1(V ) = U ∩X , denoted by V.

With these notations understood, we have DUV ⊂ Dnp as an open subset. For x ∈ Vwe have φ(x) ∈ Rp × {0}. If we write φ(x) = (φ1(x), 0), then

φ1 : V → V ⊂ Rp

is a diffeomorphism. Define a function

(2.12) φ : DUV → DUVby setting φ(v, ξ, 0) = (φ1(v), dNφv(ξ), 0) and φ(u, t) = (φ(u), t) for t 6= 0. HeredNφv : Nv → Rq is the normal component of the derivative dφv for v ∈ V. It isclear that φ is also a bijection. In particular, it induces a C∞ structure on DUV .Now, let us consider an atlas {(Uα, φα)}α∈∆ of M consisting of X−slices. Then the

collection {(DUαVα , φα)}α∈∆ is a C∞-atlas of DMX (Proposition 3.1 in [4]).

Definition 2.13 (Deformation to the normal cone). Let X ⊂M be as above. Theset DMX equipped with the C∞ structure induced by the atlas of X-slices is calledthe deformation to the normal cone associated to the embedding X ⊂M .

One important feature about the deformation to the normal cone is the functo-riality. More explicitly, let f : (M,X)→ (M ′, X ′) be a C∞-map f : M →M ′ with

f(X) ⊂ X ′. Define D(f) : DMX → DM′

X′ by the following formulas:

1) D(f)(m, t) = (f(m), t) for t 6= 0,2) D(f)(x, ξ, 0) = (f(x), dNfx(ξ), 0), where dNfx is by definition the map

(NMX )x

dNfx−→ (NM ′

X′ )f(x)

induced by TxMdfx−→ Tf(x)M

′.

Then D(f) : DMX → DM′

X′ is a C∞-map (Proposition 3.4 in [4]). In the language ofcategories, the deformation to the normal cone construction defines a functor

(2.14) D : C∞2 −→ C∞,where C∞ is the category of C∞-manifolds and C∞2 is the category of pairs ofC∞-manifolds.

Given an immersion of Lie groupoids G1ϕ→ G2, let GN1 = NG2

G1be the total space

of the normal bundle to ϕ, and (G(0)1 )N be the total space of the normal bundle to

ϕ0 : G(0)1 → G

(0)2 . Consider GN1 ⇒ (G

(0)1 )N with the following structure maps: The

source map is the derivation in the normal direction dNs : GN1 → (G(0)1 )N of the

source map (seen as a pair of maps) s : (G2, G1) → (G(0)2 , G

(0)1 ) and similarly for

the target map.The groupoid GN1 may fail to inherit a Lie groupoid structure (see counterex-

ample just before section IV in [8]). A sufficient condition is when (G(0)1 )N is a

GN1 -vector bundle over G(0)1 . This is the case when Gx1 → G

ϕ(x)2 is etale for every

x ∈ G(0)1 (in particular if the groupoids are etale) or when one considers a manifold

with two foliations F1 ⊂ F2 and the induced immersion (again 3.1, 3.19 in [8]).


The deformation to the normal bundle construction allows us to consider a C∞

structure on

Gϕ :=(GN1 × {0}

)⊔(G2 × R∗) ,

such that GN1 ×{0} is a closed saturated submanifold and so G2×R∗ is an open sub-manifold. The following results are an immediate consequence of the functorialityof the deformation to the normal cone construction.

Proposition 2.15 (Hilsum-Skandalis, 3.1, 3.19 [8]). Consider an immersion G1ϕ→

G2 as above for which (G1)N inherits a Lie groupoid structure. Let Gϕ0:=((G

(0)1 )N×

{0})⊔ (

G(0)2 × R∗

)be the deformation to the normal cone of the pair (G

(0)2 , G

(0)1 ).

The groupoid

(2.16) Gϕ ⇒ Gϕ0

with structure maps compatible with the ones of the groupoids G2 ⇒ G(0)2 and

GN1 ⇒ (G(0)1 )N , is a Lie groupoid with C∞-structures coming from the deformation

to the normal cone.

One of the interest of these kind of groupoids is to be able to define familyindices. First we recall the following elementary result.

Proposition 2.17. Given an immersion of Lie groupoids G1ϕ→ G2 as above and a

twisting α on G2. There is a canonical twisting αϕ on the Lie groupoid Gϕ ⇒ Gϕ0,

extending the pull-back twisting on G2 × R∗ from α.

Proof. The proof is a simple application of the functoriality of the deformation tothe normal cone construction. Indeed, the twisting α on G2 induces by pullback (orcomposition of cocycles) a twisting α◦ϕ on G1. The twisting α on G2 is given by aPU(H)-principal bundle Pα with a compatible left action of G2, and by definition

the twisting α ◦ ϕ on G1 is given by the pullback of Pα by ϕ0 : G(0)1 → G

(0)2 . In

particular, Pα◦ϕ = G(0)1 ×G(0)

2Pα Hence the action map G2 ×G(0)

2Pα → Pα can be

considered as an application in the category of pairs:

(G2 ×G(0)2Pα, G1 ×G(0)

1Pα◦ϕ) −→ (G

(0)2 ×G(0)

2Pα, G

(0)1 ×G(0)

1Pα◦ϕ).

We can then apply the deformation to the normal cone functor to obtain the de-sire PU(H)-principal bundle with a compatible Gϕ-action, which gives the desiredtwisting. �

3. Twisted equivariant K-theory

The crucial diference to [3] is the use of graded Fredholm bundles, which areneeded for the definition of the multiplicative structure.

Let H be a separable Hilbert space and

U(H) := {U : H → H | U ◦ U∗ = U∗ ◦ U = Id}the group of unitary operators acting on H. Let End(H) denote the space ofendomorphisms of the Hilbert space and endow End(H)c.o. with the compact opentopology. Consider the inclusion

U(H)→ End(H)c.o. × End(H)c.o.

U 7→ (U,U−1)

and induce on U(H) the subspace topology. Denote the space of unitary operatorswith this induced topology by U(H)c.o. and note that this is different from the usualcompact open topology on U(H). Let U(H)c.g be the compactly generated topology


associated to the compact open topology, and topologize the group PU(H) fromthe exact sequence

1→ S1 → U(H)c.g. → PU(H)→ 1.

Definition 3.1. Let H be a separable Hilbert space. The space Fred′(H) consist ofpairs (A,B) of bounded operators on H such that AB− 1 and BA− 1 are compactoperators. Endow Fred′(H) with the topology induced by the embedding

Fred′(H) → B(H)× B(H)× K(H)× K(H)

(A,B) 7→ (A,B,AB − 1, BA− 1)

where B(H) denotes the bounded operators on H with the compact open topologyand K(H) denotes the compact operators with the norm topology.

We denote by H = H⊕H to a Z2-graded, infinite dimensional Hilbert space.

Definition 3.2. Let U(H)c.g. be the group of even, unitary operators on the Hilbert

space H which are of the form (u1 00 u2

),

where ui denotes a unitary operator in the compactly generated topology definedas before.

We denote by PU(H) the group U(H)c.g./S1 and recall the central extension

1→ S1 → U(H)→ PU(H)→ 1

Definition 3.3. The space Fred′′(H) is the space of pairs (A, B) of self-adjoint,

bounded operators of degree 1 defined on H such that AB − I and BA − I arecompact.

Given a Z/2-graded Hilbert space H, the space Fred′′(H) is homeomorphic toFred′(H).

Definition 3.4. We denote by Fred(0)(H) the space of self-adjoint degree 1 Fred-

holm operators A in H such that A2 differs from the identity by a compact operator,with the topology coming from the embedding A 7→ (A,A2 − I) in B(H)×K(H).

The following result was proved in [1], Proposition 3.1 :

Proposition 3.5. The space Fred(0)(H) is a deformation retract of Fred′′(H).

The above discussion can be concluded telling that Fred(0)(H) is a representing

space for K-theory. The group U(H)c.g. of degree 0 unitary operators on H with

the compactly generated topology acts continuously by conjugation on Fred(0)(H),

therefore the group PU(H) acts continuously on Fred(0)(H) by conjugation. In [3]twisted K-theory for proper actions of discrete groups was defined using the rep-resenting space Fred′(H), but in order to have multiplicative structure we proceed

using Fred(0)(H).Let us choose the operator

I =

(0 II 0

).

as the base point in Fred(0)(H).

Choosing the identity as a base point on the space Fred′(H), gives a diagram of

pointed maps


Fred(0)(H)i// Fred

′′(H)

r

��

f// Fred

′(H)

Fred(0)(H)

,

where i denotes the inclusion, r is a strong deformation retract and f is a homeo-morphism. Moreover, the maps are compatible with the conjugation actions of the

groups U(H)c.g., U(H)c.g. and the map U(H)c.g. → U(H)c.g..Let X be a proper G-space and let P → X be a projective unitary G-equivariant

bundle over X. Denote by P the projective unitary bundle obtained by performing

the tensor product with the trivial bundle P(H), P = P ⊗ P(H).The space of Fredholm operators is endowed with a continuous right action of

the group PU(H) by conjugation, therefore we can take the associated bundle overX

Fred(0)(P ) := P ×PU(H) Fred(0)(H),

and with the induced G action given by

g · [(λ,A))] := [(gλ,A)]

for g in G, λ in P and A in Fred(0)(H).Denote by

Γ(X; Fred(0)(P ))

the space of sections of the bundle Fred(0)(P ) → X and choose as base point in

this space the section which chooses the base point I on the fibers. This section

exists because the PU(H) action on I is trivial, and therefore

X ∼= P /PU(H) ∼= P ×PU(H) {I} ⊂ Fred(0)(P );

let us denote this section by s.

Definition 3.6. Let X be a connected proper G-space and P a projective unitaryG-equivariant bundle over X. The Twisted G-equivariant K-theory groups of Xtwisted by P are defined as the homotopy groups of the G-equivariant sections

K−pG (X;P ) := πp

(Γ(X; Fred(0)(P ))XoG, s

)where the base point s = I is the section previously constructed.

3.1. Additive structure. There exists a natural map

Γ(X; Fred(0)(P ))XoG × Γ(X; Fred(0)(P ))XoG → Γ(X; Fred(0)(P ))XoG,

inducing an abelian group structure on the twisted equivariant K- theory groups,which we will define below. Consider for this the following commutative diagram.

Fred(0)(H)× Fred(0)(H)

��

f◦i// Fred

′(H)× Fred

′(H)

◦��

Fred(0)(H) Fred′(H)

f−1◦roo

where the vertical map denotes composition. As the maps involved in the diagram

are compatible with the conjugation actions of the groups U(H)c.g, respectivelyU(H)c.g and G, for any projective unitary G-equivariant bundle P , this induces apointed map


Γ(X; Fred(0)(P ))XoG, s)× (Γ(X; Fred(0)(P ))XoG, s)→ (Γ(X; Fred(0)(P ))XoG, s).

Which defines an additive structure in K−pG (X;P ).

3.2. Multiplicative structure. We define an associative product on twisted K-theory.

K−pG (X;P )×K−qG (X;P ′)→ K−(p+q)G (X;P ⊗ P ′)

Induced by the map

(A,A′) 7→ A⊗I + I⊗A′

defined in Fred0(H), and ⊗ denotes the graded tensor product, see [2] in pages24-25 for more details. We denote this product by •.

Definition 3.7 (Equivariant Total Twisted K-theory). For every p ∈ N, the degreep, G-equivariant Total Twisted K-theory group is given by

(3.8) K−pTW,G(X) :=⊕

α∈H1(G,PU(H))

K−pG (X,Pα).

By the discussion above

(3.9) KevTW,G(X) :=

⊕p∈N, even

K−pTW,G(X)

has a ring structure and

(3.10) KoddTW,G(X) :=

⊕p∈N, odd

K−pTW,G(X)

is a KevTW,G(X)-module. The last groups above are called the even, respectively

odd, G−equivariant Total Twisted K-theory groups of X.

3.3. Topologies on Fredholm Operators. In [16] a Fredholm picture of twistedK-theory is introduced. Denote by Fred′(H)s∗ the space whose elements are thesame as Fred′(H) but with the strong ∗-topology on B(H).

Definition 3.11. [16, Thm. 3.15] Let X be a connected G-proper space and Pa projective unitary G-equivariant bundle over X. The Twisted G-equivariant K-theory groups of X (in the sense of Tu-Xu-Laurent) twisted by P are defined as thehomotopy groups of the G-equivariant strong∗-continuous sections

K−pG (X;P ) := πp(Γ(X; Fred′(P )s∗)

G, s).

The bundle Fred′(P )s∗ is defined in a similar way as Fred′(P ).

We will prove that the functors K∗G(−, P ) and K∗G(−, P ) are naturally equivalent.

Lemma 3.12. The spaces Fred′(H) and Fred′(H)s∗ are PU(H)-weakly homotopyequivalent.

Proof. The strategy is to prove that Fred′(H)s∗ is a representing of equivariantK-theory. The same proof for Fred′(H) in [1, Prop. A.22] applies. In particularGL(H)s∗ is G-contractible because the homotopy ht constructed in [1, Prop. A.21]is continuous in the strong∗-topology and then the proof applies. �

Using the above lemma one can prove that the identity map defines an equiv-alence between (twisted) cohomology theories K∗G(−, P ) and K∗G(−, P ). Then wehave that the both definitions of twisted K-theory are equivalents. Summarizing


Proposition 3.13. For every proper G-manifold X and every projective unitaryG-equivariant bundle over X. We have an isomorphism

K−pG (X;P ) ∼= K−pG (X;P ).

Remark 3.14. In order to simplify the notation from now on we denote by H to

a Z2-graded separable Hilbert space and we denote by Fred(0)(P ) to the bundle

Fred(0)(P ).

3.4. Relation with the Kasparov external product. In [16] twisted K-theoryfor Lie groupoids is defined and in Prop. 6.11 of that work this group is describedas a KK-group for the case of proper groupoids.

Proposition 3.15. [16, Prop. 6.11] If G⇒M is a proper Lie groupoid andM/G iscompact, then for i = 0, 1, there is a natural isomorphism χ : KKi

G(C0(M), BP )→KiG(M,P ), where BP is certain C∗-algebra associated to the twisting P .

Using the external Kasparov product they define a product

KiG(M,P )⊗Kj

G(M,P ′)•TXL

// Ki+jG (M,P ⊗ P ′).

Using the functoriality of both products • and •TXL one can prove that they arethe same.

Definition 3.16. (i) If Φ is a KKG(A,B)-cycle we denote by Φ∗ to the ho-momorphism

Φ∗ : KKG(C0(M), B0)→ KKG(C0(M), BP )

x 7→ x •TXL Φ.

(ii) If s ∈ ΓG(Fred(0)(P )) we denote by s∗ the homomorphism

s∗ : KiG(X)→ Ki

G(X,P )

[f ] 7→ [s • f ].

Proposition 3.17. If Φ ∈ KKiG(C0(M), BP ) and Ψ ∈ KKi

G(C0(M), BP ′), thenχ(Φ •tuxustacks Ψ) = χ(Φ) • χ(Ψ).

Proof.

χ(Φ •TXL Ψ) = χ(Φ∗(1C0(M)) •TXL Ψ∗(1C0(M)))

= χ(Φ∗(Ψ∗(1C0(M))))

= (χ(Φ))∗((χ(Ψ))∗(χ(1C0(M)))

= (χ(Φ))∗(1M ) • (χ(Ψ))∗(1M )

= (χ(Φ)) • (χ(Ψ)).

�

The above result implies that both products are the same module the equivalenceχ.

4. Thom isomorphism

Let G ⇒ G0 be a Lie groupoid and P a twisting. Consider a G-oriented vectorbundle E −→ X. In particular since we will assume that G acts properly on Pand on E, we can assume E admits a G-invariant metric, see for instance [14]proposition 3.14 and [7] theorem 4.3.4.. As explained in [5] appendix A (speciallyproposition A.3), in this situation there is a natural isomorphism

Th : K∗G(X,P)→ K∗−rank(E)G (E, π∗(P⊗ βE))


where K∗G(X,P) stands for the K-theory of the twisted groupoid C∗-algebra C∗r (XoG,P ) and where βE is the orientation G-twisting over E defined in example ((ii))in 2.9 above. The fact that it is indeed the Thom isomorphism comes from thefunctoriality and the naturality with respect to the Kasparov products of the LeGall’s descent construction [10] theorem 7.2. This is explained in details in the ap-pendix cited above or in [12] in the context of real groupoids (the same argumentsapply in the complex case).

Now, in [16] theorem 3.14 the authors prove that for proper Lie groupoids the

groups K∗G(X,P) and K−pG (X;P ) are naturally isomorphic. We thus obtain, byproposition 3.13, the Thom isomorphism

Th : K∗G(X,P )→ K∗−rank(E)G (E, π∗(P ⊗ βE)).

It is possible however to construct the Thom isomorphism directly in the Fred-holm picture of the twisted K-theory (whenever the respectives action groupoidsare proper), we will perform this construction for the benefit of the reader.

The spin representation and twisted K-Theory. Let n be an even naturalnumber.

Let Rn denote the euclidean, n-dimensional vector space denoted with the eu-clidean scalar product.

The Clifford algebra Cliff(Rn) is defined as the complexification of the quotient

of the tensor algebra TRn =∞⊗j=0

Rn by the two-sided ideal defined by elents of the

form x⊗ x− 〈x, x〉, where 〈〉 denotes the euclidean scalar product.It is generated as C-algebra by the elements of a an orthogonal basis ei of Rn

with the relations ei · ej = −2δi,j .The algebra Cliff(Rn) is isomorphic as a vector space to the exterior algebra

Λ∗(Rn) =n⊕j=0

ΛjRn [9], Proposition 1.3 in page 10, in particular, it has complex

dimension 2n .The map given by Clifford multiplication with the element e1·, . . . , ·en defines

a linear operator on Cliff(Rn). The Clifford algebra then decomposes as a vectorspace Cliff(Rn) = S+ ⊕ S−, where S+ is the eigenspace associated to +1 and S−

is the one associated with −1. An element in S+ is called even, an element in S−

is said to be odd.The group Spin(Rn) consists of the multiplicative group of even units in the

Clifford algebra, in symbols Spin(Rn) = Cliff(Rn)∗ ∩ S+.The group Spin(Rn) is the universal covering of the special orthogonal group

SO(n). The map

1→ Z2 → Spin(Rn)→ SO(n)→ 1

is a model for the universal central extension of SO(n).This extension is classified by the nontrivial class τ ∈ H2(SO(n), S1) ∼= Z2.The group Spin(Rn) has a complex linear representation ρ : Spin(Rn)→ U(2n),

given by the identification of Cliff(Rn) = Cliff(Rn) ⊗ C with the complex vectorspace of dimension 2n as an algebra, and the linear operator given by ρ(x) : v 7→x−1vx.


The representation ρ gives rise to a continuous group homomorphism β as in thefollowing diagram:

1 // S1 //

��

// Spinc(Rn) //

ρ

��

SO(n) //

β

��

1

1 // S1 // U(2n) // PU(2n) // 1

Definition 4.1. The spin representation is the homomorphism β : SO(n) →PU(H)

Remark 4.2. Let n be a even positive integer. Consider a proper oriented G-

vector bundle Eπ−→ X over a proper G-manifold X. We can suppose that the chart

data is given by a generalized morphism

X oGOE// SO(n) .

Composing the generalized morphism OE with the spin representation β we obtain

a twisting βE : X oG // PU(H) , called the orientation twisting.

We will construct now the Thom class in the Fredholm picture. If X is a properG-manifold, by Theorem 2.3 in [17] for every x ∈ X there is a open neighbourhoodU of x contractible to the orbit of x in X oG with action of the isotropy group Gxsuch that there is a Lie groupoid isomorphism

(X oG) | U ∼= U oGx.

We have an isomorphism

(4.3) K−nGx (U, βE |U ) ∼= RS1(Gx),

where Gx is the S1-central extension of Gx associated to the twisting βE |U . Onthe other hand, E |{x} is a real representation of Gx, since it can be viewed as ahomomorphism ηx : Gx → SO(n). The composition β ◦ ηx : G→ PU(H) is a pro-

jective representation and its isomorphism class determines an element of RS1(Gx).Using the identification 4.3, it can be viewed as an element of K−nGx (U, βE |U ). We

denote this element by λU−1.Taking a covering of X oG one can see that these local elements are the same

on intersections. The local trivializations define a global element

[λE−1] ∈ K−nG (X,βE),

we call it the Thom class.Given s ∈ ΓG(P ×PU(H) Fred(0)(H)), where P → X is a twisting, we define the

Thom isomorphism

Th : K∗G(X,P )→K∗−nG (E, π∗(P ⊗ βE))

[s] 7→[e 7→ s(π(e)) • λE−1(π(e))].

When the vector bundle E is odd dimensional, using the classic suspension iso-morphism and the previous Thom isomorphism for E⊕R, one gets as well a Thomisomorphism as above.

Since the Thom isomorphism is natural with respect to the Kasparov productwe can resume the discussion above in the following statement.

Theorem 4.4. [Thom isomorphism] With notations as above, there is a naturalisomorphism

Th : K∗G(X,P )→ K∗−rank(E)G (E, π∗(P ⊗ βE))


which gives the Thom isomorphism.

(i) If E is a Spinc G− vector bundle of even ranck, then βE is trivial and oneobtains a ring isomorphism

Th : KevTW,G(X)→ Kev

TW,G(E)

and an isomorphism of KevTW,G(X)-modules

Th : KoddTW,G(X)→ Kodd

TW,G(E).

(ii) If E is a Spinc G− vector bundle of odd rank, then βE is trivial and oneobtains a Kev

TW,G(X)-module isomorphism

Th : KevTW,G(X)→ Kodd

TW,G(E)

and a KevTW,G(X)-module isomorphism

Th : KoddTW,G(X)→ Kev

TW,G(E).

5. Pushforward Map

In this section we will recall how to define the pushforward morphism associatedto any smooth G-map f : X → Y between to G- manifolds, definition 4.1 in [5]. Forthe purpose of this paper we will perform the construction in the case of K-orientedmaps. By this we mean that the bundle T ∗X⊕ f∗(TY ) admits a Spinc−structure.

The difference in the present construction with respect to ref.cit. is that wewill not make reference to C∗-algebras and we will perform the construction usingthe Fredholm picture of the twisted K-theory, in particular the construction belowworks only for G-proper manifolds.

We will need to state some general statements about groupoids that will simplifythe particular constructions we are interested in.

Lemma 5.1. Let G⇒ G0 be a proper Lie groupoid together with a twisting P. LetH ⇒ H0 be a proper Lie saturated closed subgroupoid.

(i) There is a canonical restriction morphism

(5.2) K−pG (G0, P )→ K−pH (H0, P |H0)

(ii) Suppose G decomposes as the union of two saturated proper subgroupoidsG = H tH ′ ⇒ H0 tH ′0 with H closed subgroupoid. There is a long exactsequence

(5.3)

// K−pH′ (H

′0, P |H′0

) // K−pG

(G0, P ) // K−pH

(H0, P |H0) // K

−p−1

H′ (H′0, P |H′0) //

Lemma 5.4. Let G ⇒ G0 be a proper Lie groupoid together with a twisting P,consider the product groupoid G × (0, 1] ⇒ G0 × (0, 1] with the pullback twistingP(0,1]. For every p ∈ Z

K−pG×(0,1](G0 × (0, 1], P(0,1]) = 0.

The two previous lemmas are classic in the C∗-algebraic context, i.e., once weuse that the isomorphism between the twisted K-theory with the C∗-picture andthe twisted K-theory with the Fredholm picture (theorem 3.14 [16]).

The following result is an immediate consequence of lemmas 5.1 and 5.4 above.

Proposition 5.5. Given an immersion of proper Lie groupoids G1ϕ→ G2 and a

twisting α on G2, consider the twisted deformation groupoid (Gϕ, Pα) of section 2.3(propositions 2.15 and 2.17). The morphism in K-theory induced by the restrictionat zero,

(5.6) K−pGϕ(G(0)ϕ , Pϕ)

e0// K−pG2

(G(0)2 , P2)


is an isomorphism.

Definition 5.7 (Index associated to a groupoid immersion). Given an immersion

of proper Lie groupoids G1ϕ→ G2 as above and a twisting α on G2, we let

(5.8) Indϕ : K−pGN1

((G(0)1 )N , PN1 )→ K−pG2

(G(0)2 , P2)

to be the morphism in K-theory given by Indϕ := e1 ◦ e−10 .

We are ready to define the shriek map. Let G⇒ G0 be a Lie groupoid togetherwith a twisting P . Let X,Y be two G-proper manifolds and let f : X → Y be asmooth G-map with T ∗X ⊕ f∗TY a G-Spinc vector bundle that we will assume ina first time to have even rank. We will also assume the moment maps X → G0

and Y → G0 to be submersions, then T ∗X ⊕ f∗TY being Spinc is equivalent toVf := T ∗vX ⊕ f∗TvY being Spinc. The shriek morphism

(5.9) f ! : K−pG (X,PX)f!// K−p−dfG (Y, PY ),

where df := rank Vf , will be given as the composition of the following three mor-phism

I. The twisted G-equivariant Thom isomorphism

(5.10) K−pG (X,PX)T∼=// K−p−dfG (T ∗vX

⊕f∗TvY, PVf ).

II. We consider now the index morphism

(5.11) K−p−df(TvX

⊕f∗TvY )oG(f∗TvY, P )

Ind// K−p−dff∗TvYo(TvXoG)(f

∗TvY, P )

associated to the immersion

f∗TvY oG −→ f∗TvY o (TvX oG)

given by the product of the identity in G and the inclusion of the units f∗TvY inthe groupoid f∗TvY o TvX.

III. Consider the groupoid immersion

(5.12) X oGf

// (Y ×G0(X ×G0

X)) oG,

where f := (f ×4)× IdG. Then the induced deformation groupoid is

Gf oG

where

Gf ⇒ G(0)f

is the groupoid given by

(5.13) Gf := f∗(TvY ) o TvX × {0}⊔Y ×G0 (X ×G0 X)× (0, 1] and

(5.14) G(0)f = f∗TvY × {0}

⊔Y ×G0

X × (0, 1]

Notice that Y ×G0 (X ×G0 X) and Y are Morita equivalent groupoids with Moritaequivalence the canonical projection.

Let αf the twisting on Gf o G given by proposition 2.17. It is immediate tocheck that αf |(f∗(TvY )oTvX)oG = π∗f∗TvYoTvXα.

We can hence consider the twisted deformation index morphism associated to(Gf oG,αf ) :


(5.15) K−p−dff∗TvYo(TvXoG)(f

∗TvY, P )Indf

// K−p−df(Y×G0

(X×G0X))oG(Y ×G0

X,P )

µ∼=��

K−p−dfG (Y, P )

For composing 5.10 with 5.11 remember that by the Fourier isomorphism provedin proposition 2.12 in [6] and by theorem 3.14 in [16] we have an isomorphism

K∗G(T ∗vX⊕

f∗TvY, PVf ) ≈ K∗(TvX⊕f∗TvY )oG(f∗TvY, P ).

We can now give the following definition:

Definition 5.16 (Pushforward morphism for twisted G-manifolds). Let X,Y betwo manifolds and f : X −→ Y a smooth map. Under the presence of a twisting Pon G we let

(5.17) K−pG (X,PX)f!// K−p−dfG (Y, PY )

to be the morphism given by the composition of the three morphisms describedabove, 5.10 followed by 5.11 followed by 5.15.

One of the main results is that the pushforward maps induce a ring morphismsbetween the total twisted K-theory rings:

Proposition 5.18. Let G ⇒ G0 be a Lie groupoid. Let X,Y be two G-propermanifolds and let f : X → Y be a G-smooth K-oriented map. We have that

(i) If T ∗vX ⊕ f∗TvY has even rank, the pushforward map

f ! : KevTW,G(X)→ Kev

TW,G(Y )

is a ring morphism and the pushforward map

f ! : KoddTW,G(X)→ Kodd

TW,G(Y )

is a KevTW,G(X)-module morphism.

(ii) If T ∗vX ⊕ f∗TvY has odd rank, the pushforward map

f ! : KevTW,G(X)→ Kodd

TW,G(Y )

is KevTW,G(X)-module morphism and the pushforward map

f ! : KoddTW,G(X)→ Kodd

TW,G(Y )

is a KevTW,G(X)-module morphism.

Proof. By definition f ! is constructed by means of a Thom isomorphism and oftwo deformation indices. These indices are at their turn constructed by restriction(or evaluation) morphisms. To conclude the proof one has only to observe thatrestrictions induce ring/modules morphisms together with the fact that Thom is aring/module isomorphism, see 4.4. �

The main theorem in [5] (theorem 4.2) can be now written as follows

Theorem 5.19. The above push-forward morphism is functorial, that means, ifwe have a composition of smooth K-oriented G-maps between G− proper manifoldswith T ∗vX ⊕ f∗TvY and T ∗v Y ⊕ f∗TvZ of even rank:

(5.20) Xf−→ Y

g−→ Z,


Then the following diagram of ring morphisms commutes

KevTW,G(X)

(g◦f)!//

f!&&

KevTW,G(Z)

KevTW,G(Y )

g!

88

In the case the bundles T ∗vX ⊕ f∗TvY and T ∗v Y ⊕ f∗TvZ are not both simulta-neously of even rank a similar conclusion is obtained for modules morphisms.

6. Connes Approach to twisted K Homology for Lie groupoids

The pushforward functoriality theorem (thm. 4.2 in [5]) allows us to give thefollowing definition:

Definition 6.1 (Twisted geometric K-homology). Let G ⇒ M be a Lie groupoidwith a twisting P . By the ”Twisted geometric K-homology group” associated to(G,P ) we mean the abelian group denoted by Kgeo

∗ (G,P ) with generators the cycles(X,x) where

(1) X is a smooth co-compact G-proper manifold,(2) πX : X → M is the smooth momentum map which supposed to be a

K-oriented submersion and(3) x ∈ K−pG (X,PX) for some p ∈ N,

and relations given by

(6.2) (X,x) ∼ (X ′, g!(x))

where g : X → X ′ is a smooth G-equivariant map.The group defined above admits a Z2-gradation

Kgeo∗ (G,P ) = Kgeo

0 (G,P )⊕

Kgeo1 (G,P ).

where Kgeoj (G,P ) is the subgroup generated by cycles (X,x) for which TvX has

rank congruent to j modulo 2.

We will now describe a product between two cycles with possibly different twist-ings by using the product structure defined in previous sections. For that we willneed first to recall the a basic operation:

The Pullback: Let Ah−→ B be a smooth G-equivariant map (A,B G-proper

manifolds). Suppose we have a twisting P on G. We are going to consider, forevery q ∈ N, the pullback

(6.3) h∗ : K−qG (B,PB) −→ K−qG (A,PA)

given as follows: If γ : Sq → Γ(B,Fred(PB))G is a continuous map with γ(∗) = sone let

h∗γ : Sq → Γ(A,Fred(PA))G

to be given by(h∗γ)(z)(a) := γ(z)(h(a)),

it is then classic to show that it induces a map between the homopoty classes.More generally we will need a pullback map associated to aG-equivariant Hilsum-

Skandalis map. We explain next what do we mean by this.Consider a Lie groupoid HA ⇒ A, we say that it is a G-groupoid if G acts on

HA, on A and the source and target maps of HA are G−equivariant. Under thissituation we might form the semi-direct product groupoid

HA oG⇒ A.


Suppose now that we have two G-proper (all the actions are required to be proper)Lie groupoids HA and HB together with a generalized morphism h : HA−−− > HB

between them, that is, suppose we are given a HB-principal bundle Ph over HA,putting this in a diagram:

HA

��

Phsh

!!

th

}}

HB

��

A B

We are going to consider, for every q ∈ N, the pullback

(6.4) h∗ : K−qHBoG(B,PB) −→ K−qHAoG(A,PA)

given as follows: If γ : Sq → Γ(B,Fred(PB))HBoG is a continuous map withγ(∗) = s one let

h∗γ : Sq → Γ(A,Fred(PA))HAoG

to be given by(h∗γ)(z)(a) := γ(z)(b)

where b = sh(v) for some v ∈ t−1h (a). One proves using the invariance of γ together

with the identification Ph×HBoG Fred(PB) = Fred(PA) that the definition of h∗γdoes not depend on the choice of v.

Lemma 6.5. The pullback is natural. The following properties hold:

(i) Id∗ = Id(ii) (h2 ◦ h1)∗ = h∗1 ◦ h∗2

Remark 6.6 (On Le Gall’s descent functors). The definition of the pullback aboverecalls Le Gall’s pullback construction on the untwisted case which generalizesKasparov descent morphisms. The simplicity of our construction is due to thefact that we are only dealing with the proper action case. In the general case iscertainly possible to adapt Le Gall’s to S1-central extensions and then to apply itto the Twisted K-theory case. We do not need to do it in this generality in thispaper.

The main property is the naturality of the pushforward maps with respect topullbacks, this is the new and the main key technical result in this paper.

Proposition 6.7. Let G ⇒ G0 be a Lie groupoid together with a twisting P .Suppose we have a commutative diagram of G-smooth K-oriented maps betweenG-proper manifolds

Ag//

p

��

A′

q

��

Bf// B′

Then we have the following equality between K-theory morphisms

g! ◦ p∗ = q∗ ◦ f !

Proof. We have to show that the following diagram is commutative

(6.8) K∗G(A,PA)g!// K∗G(A′, PA′)

K∗G(B,PB)

p∗

OO

f !// K∗G(B′, PB′)

q∗

OO


We will split the above diagram in four commutative diagrams:Diagram I. Consider the following commutative diagram of groupoid mor-

phisms which are equivariant with respect to the G-action:

A′ ×G0 (A×G0 A)

q×4(p)

��

IdA′×PrG0// A′ ×G0 G0

q×IdG0

��

B′ ×G0(B ×G0

B)IdB′×PrG0

// B′ ×G0G0

Once identifying A′ ×G0G0 with A′ (and respectively for B′) we have that IdA′ ×

PrG0induces the Morita equivalence of groupoids between A′ ×G0

(A×G0A) and

A′ with inverse a Hilsum-Skandalis isomorphism that induces the isomorphism µin K-theory. Hence the diagram above induces the following commutative diagramin K-theory:

(6.9) K∗Gn(A′×G0(A×G0

A))(A′ ×G0

A,PA′×G0A)

µ

≈//

I

K∗G(A′, PA′)

K∗Gn(B′×G0(B×G0

B))(B′ ×G0

B,PB′×G0B)

(q×G04(p))∗

OO

µ

≈// K∗G(B′, PB′)

q∗

OO

Diagram II. Remember the G-groupoid immersions

Ag×4−→ A′ ×G0

(A×G0A)

and

Bf×4−→ B′ ×G0 (B ×G0 B)

used above to construct the deformation indices ( see (5.12) and 5.15). They fit inthe following commutative diagram of G-morphisms:

(6.10) A′ ×G0(A×G0

A)q×4(p)

// B′ ×G0(B ×G0

B)

A

g×4

OO

p// B

f×4

OO

By the functoriality of the deformation to the normal cone we have a morphismof G-groupoids (see (5.13) for notations)

Gg

��

˜q×4(p)// Gf

��

G(0)g

( ˜q×4(p))0

// G(0)f

whose restriction at t = 1 gives q × 4(p) and whose restriction at t = 0 givesdvp n dvq : TvA n g∗TvA

′ → TvB n f∗TvB′ as a morphism of G-groupoids where

dvp (resp. dvq) stands for the derivative in the tangent vertical direction. Sincepullbacks obviuosly commutes with restrictions we have the following commutative


diagram(6.11)

K∗Gn(TvAng∗TvA′)(g∗TvA

′, Pg∗TvA′)Indg

//

II

K∗Gn(A′×G0(A×G0

A))(A′ ×G0

A,PA′×G0A)

K∗Gn(TvBnf∗TvB′)(f∗TvB

′, Pf∗TvB′) Indf

//

(dvpndvq)∗OO

K∗Gn(B′×G0(B×G0

B))(B′ ×G0

B,PB′×G0B)

(q×G04(p))∗

OO

Diagram III. The groupoid morphism (equivariant w.r. to G)

dvpn dvq : TvAn g∗TvA′ → TvB n f∗TvB

′

induces (again by functoriality of the deformation to the normal cone) a G-groupoidmorphism between the respective tangent groupoids

(dvpn dvq)tan : (TvAn g∗TvA′)tan → (TvB n f∗TvB

′)tan

whose restriction at t = 1 gives dvp n dvq and whose restriction at zero givesdvp⊕ dvq : TvA⊕ g∗TvA′ → TvB ⊕ f∗TvB′. For the same reason as diagram II wehave the following commutative diagram in K-theory:

(6.12)

K∗Gn(TvA⊕g∗TvA′)(g∗TvA

′, Pg∗TvA′)

III

Ind// K∗Gn(TvAng∗TvA′)(g

∗TvA′, Pg∗TvA′)

K∗Gn(TvB⊕f∗TvB′)(f∗TvB

′, Pf∗TvB′)

(dvp⊕dvq)∗OO

Ind// K∗Gn(TvBnf∗TvB′)(f

∗TvB′, Pf∗TvB′)

(dvpndvq)∗OO

Diagram IV. The commutativity of the following diagram follows from thenaturality of Thom isomorphism:

(6.13) K∗G(A,PA)Thom

≈// K∗Gn(TvA⊕g∗TvA′)(g

∗TvA′, Pg∗TvA′)

K∗G(B,PB)

p∗

OO

Thom

≈// K∗Gn(TvB⊕f∗TvB′)(f

∗TvB′, Pf∗TvB′)

(dvp⊕dvq)∗OO

By definition, diagram (6.8) decomposes, with the previous diagrams, in thefollowing form:

//

IV

//

III

//

II

//

I

//

OO OO

//

OO

//

OO

//

OO

and hence it is commutative. �

The product of two cycles: Let P and Q two twistings on G. Let (X,x) with

x ∈ K−pG (X,PX) and (Y, y) with y ∈ K−qG (Y,QY ) we put(6.14)

(X,x) · (Y, y) := (X ×G0Y, π∗Xx · π∗Y y) ∈ K−p−qG (X ×G0

Y, PX×G0Y ⊗QX×G0

Y )

where πX , πY stand for the respective projections from X ×G0 Y to X and Y .The following is the main result of this paper:

Theorem 6.15. For any Lie groupoid, the Total twisted geometric K-homologygroup

KgeoTW,0(G) :=

⊕α∈H1(G,PU(H))

Kgeo0 (G,Pα)


has a ring structure with the product described above and

KgeoTW,1(G) :=

⊕α∈H1(G,PU(H))

Kgeo1 (G,Pα)

has a KgeoTW,0(G)-module structure.

Proof. We only have to prove that the product described above is well definedin the K-homology group. Let P and Q two twistings on G. Let (X,x) with

x ∈ K−pG (X,PX) and (Y, y) with y ∈ K−qG (Y,QY ). Suppose we have smooth maps

Xg−→ X ′ and Y

f−→ Y ′. We would finish if we can show that

(X ×G0Y, π∗Xx · π∗Y y) ∼ (X ′ ×G0

Y ′, π∗X′g!x · π∗Y ′f !y).

In fact we can consider the smooth map

X ×G0Y

g×f−→ X ′ ×G0Y ′

which fits the following commutative diagrams

X ×G0Y

πX

��

g×f// X ′ ×G0

Y ′

πX′

��

Xg

// X ′

and

X ×G0Y

πY

��

g×f// X ′ ×G0

Y ′

πY ′

��

Yf

// Y ′

The result now follows from proposition 6.7 and proposition 5.18 since they imply

(g × f)!(π∗Xx · π∗Y y) = (g × f)!(π∗Xx) · (g × f)!(π∗Y y) = π∗X′g!x · π∗Y ′f !y

and hence

(X ×G0Y, π∗Xx · π∗Y y) ∼ (X ′ ×G0

Y ′, π∗X′g!x · π∗Y ′f !y).

�

7. The Baum-Connes map

Recall that in [5] the Baum-Connes assembly map

(7.1) Kgeo∗ (G,Pα)

µα// K−∗(G,Pα)

was constructed for every twisting α on G where K∗(G,Pα) := K−∗(C∗r (G,Pα))

stands for the K−theory of the reduced C∗-algebra associated to the twistedgroupoid (G,Pα), (there is also the assembly map taking values on the maximalC∗-algebra). The definition of the Baum-Connes map is given by

µα(X,x) := πX !(x) ∈ K∗(G,Pα)

where πX ! is the pushforward map defined in [5]. Consider the Total TwistedK-theory group

K∗TW (G) :=⊕

α∈H1(G,PU(H))

K∗(G,Pα).


By the theorem above we have a ring (module for the odd case) structure on theimage of the Total twisted Baum-Connes assembly map

(7.2) KgeoTW,0(G)

µTW// K0

TW (G)

where µTW := ⊕µα whenever µTW is injective. In particular if µTW is an isomor-phism then K0

TW (G) has a ring (module for the odd case) structure such that µTWis a ring isomorphism.

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Centro de Ciencias Matematicas. UNAM, Ap.Postal 61-3 Xangari. Morelia, Mi-

choacan MEXICO 58089

E-mail address: [email protected]

URL: http://www.matmor.unam.mx~barcenas

Institut de Mathematiques de Toulouse., 118, route de Narbonne F-31062 Toulouse,FRANCE


URL: http://www.math.univ-toulouse.fr/~carrillo

Centro de Ciencias Matematicas. UNAM, Ap.Postal 61-3 Xangari. Morelia, Mi-

choacan MEXICO 58089

Departamento de Matematicas., Pontificia Universidad Javeriana, Cra. 7 No. 43-82

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URL: https://sites.google.com/site/mavelasquezm/

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MULTIPLICATIVE STRUCTURES AND THE TWISTED BAUM-CONNES ...barcenas/BACAVE.pdf · group via the assembly map. This is of course the case when this twisted Baum-Connes map is an isomorphism